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| In [[mathematics]], the '''Pincherle derivative''' ''T’'' of a [[linear operator]] ''T'':'''K'''[''x''] → '''K'''[''x''] on the [[vector space]] of [[polynomial]]s in the variable ''x'' over a [[field (mathematics)|field]] '''K''' is the [[commutator]] of ''T'' with the multiplication by ''x'' in the [[endomorphism ring|algebra of endomorphisms]] End('''K'''[''x'']). That is, ''T’'' is another linear operator ''T’'':'''K'''[''x''] → '''K'''[''x'']
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| :<math> T' := [T,x] = Tx-xT = -\operatorname{ad}(x)T,\,</math>
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| so that
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| :<math> T'\{p(x)\}=T\{xp(x)\}-xT\{p(x)\}\qquad\forall p(x)\in \mathbb{K}[x].</math>
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| This concept is named after the Italian mathematician [[Salvatore Pincherle]] (1853–1936).
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| == Properties ==
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| The Pincherle derivative, like any [[commutator]], is a [[derivation (abstract algebra)|derivation]], meaning it satisfies the sum and products rules: given two [[linear operator]]s <math>\scriptstyle S </math> and <math>\scriptstyle T </math> belonging to <math> \scriptstyle \operatorname{End} \left( \mathbb K[x] \right) </math>
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| #<math>\scriptstyle{ (T + S)^\prime = T^\prime + S^\prime }</math> ;
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| #<math>\scriptstyle{ (TS)^\prime = T^\prime\!S + TS^\prime }</math> where <math>\scriptstyle{ TS = T \circ S}</math> is the [[Function_composition#Composition_operator|composition of operators]] ;
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| One also has <math>\scriptstyle{ [T,S]^\prime = [T^\prime , S] + [T, S^\prime ] }</math> where <math>\scriptstyle{ [T,S] = TS - ST}</math> is the usual [[Lie algebra|Lie bracket]], which follows from the [[Jacobi identity]].
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| The usual derivative, ''D'' = ''d''/''dx'', is an operator on polynomials. By straightforward computation, its Pincherle derivative is
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| : <math> D'= \left({d \over {dx}}\right)' = \operatorname{Id}_{\mathbb K [x]} = 1.</math>
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| This formula generalizes to
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| : <math> (D^n)'= \left({{d^n} \over {dx^n}}\right)' = nD^{n-1},</math>
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| by [[mathematical induction|induction]]. It proves that the Pincherle derivative of a [[differential operator]]
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| : <math> \partial = \sum a_n {{d^n} \over {dx^n} } = \sum a_n D^n </math>
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| is also a differential operator, so that the Pincherle derivative is a derivation of <math>\scriptstyle \operatorname{Diff}(\mathbb K [x]) </math>.
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| The shift operator
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| : <math> S_h(f)(x) = f(x+h) \, </math> | |
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| can be written as
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| : <math> S_h = \sum_{n=0} {{h^n} \over {n!} }D^n </math>
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| by the [[Taylor formula]]. Its Pincherle derivative is then
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| : <math> S_h' = \sum_{n=1} {{h^n} \over {(n-1)!} }D^{n-1} = h \cdot S_h. </math> | |
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| In other words, the shift operators are [[eigenvector]]s of the Pincherle derivative, whose spectrum is the whole space of scalars <math>\scriptstyle{ \mathbb K }</math>.
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| If ''T'' is [[shift-equivariant]], that is, if ''T'' commutes with ''S''<sub>''h''</sub> or <math>\scriptstyle{ [T,S_h] = 0}</math>, then we also have <math>\scriptstyle{ [T',S_h] = 0}</math>, so that <math>\scriptstyle T'</math> is also shift-equivariant and for the same shift <math>\scriptstyle h</math>.
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| The "discrete-time delta operator"
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| : <math> (\delta f)(x) = {{ f(x+h) - f(x) } \over h }</math>
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| is the operator
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| : <math> \delta = {1 \over h} (S_h - 1),</math>
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| whose Pincherle derivative is the shift operator <math>\scriptstyle{ \delta ' = S_h }</math>.
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| == See also ==
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| *[[Commutator]]
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| *[[Delta operator]]
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| *[[Umbral calculus]]
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| == External links ==
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| *Weisstein, Eric W. "''[http://mathworld.wolfram.com/PincherleDerivative.html Pincherle Derivative]''". From MathWorld—A Wolfram Web Resource.
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| *''[http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Pincherle.html Biography of Salvatore Pincherle]'' at the [[MacTutor History of Mathematics archive]].
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| [[Category:Differential algebra]]
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